Chapter 4 Chapter 4 State Space Solutions and Realizations Objectives: Concepts of linear Ordinary Differential Equations Solution of linear state equations Similarity Transformation Discretization of state space model Realization of rational transfer functions 。Canonical forms heory
Chapter 4 Chapter 4 State Space Solutions and Realizations Objectives: • Concepts of linear Ordinary Differential Equations • Solution of linear state equations • Similarity Transformation • Discretization of state space model • Realization of rational transfer functions • Canonical forms Chapter 4 1
Chapter 4 3 4.1 Solutions of LTI system 1.Eigenvalues,Eigenvectors Solution of Differential Equations Define a linear homogeneous differential equationsystem X(t)=AX(t) or x() a a2 an 「x) 无(t) d an … 七() … … … 元0 a a… x() Suppose the solution is in the form of x,()=c,e =1,2., Substituting the solution into the equation a11 a12. ain e"= a21a2 …a2m C an an ann Cn then we get 见=A That is AC=AC So A is a eigenvalue of matrix A and vector C is the eigenvector of A
Chapter 4 2
Chapter 4 4 Furthermore [A-IC=0 which having non-vanishing roots if and only if A-I=0 which is called the characteristic equation of differential equation system. Suppose the root vector '=[,5,,灯 whereare roots of characteristic equation. By solving equation[A-I]C=0,we obtain the eigenvactor corresponding to. Summary Suppose a dynamic system S S:X0=AX(0,X(0∈R,A∈nn) and if there are non-vanishing vector ceR and a complex number such that AC=AC then we say A is an eigenvalue of matrix A; C is an eigenvector corresponding to of matrix A. The determinant A-is called the characteristic polynomial of system S. A-=0 is the characteristic equation of system S
Chapter 4 3
Chapter 4 5 Example: If A=diag(an.a..m),then aa are the eigenvalues of the matrix. Because in this case A-2川=(a1-0(a2-2)-(am-)=0 therefore 元=aa =1,2,,) Let and c:(i=1,2,...,n)are the eigenvalues and the corresponding eigenvectors then the solution of the system X'(t)=k Cie+kCze+Cre wherekk.,k are constant coefficients which can be obtained by substituting n-boundary conditions
Chapter 4 4
Chapter 4 6 2.State Matrix If the real part of every eigenvalues Ais negative,that is Re()<o then lim X(t)=0 the system is stable,otherwise lim X(t)=co the system is unstable. Consider the linear time-invariant control system differential equation X(t)=AX(t)+BU(t) where Aeh(n,n),Beh(n,r),X(t)ER",U()ER'is a controlvector. A is called the system matrix（系统矩阵)in open-loop case. If the controllaw is U=-KX(t),then X(t)=(A-BK)X() So the condition of stability of the closed loop control system is that the real part of every eigenvalue of matrix(A-BK)is negative
Chapter 4 5